# CALCULUS 2

## Learning outcomes of the course unit

Knowledge and understanding: At the end of this course the student should know the essential definitions and results of analysis in several variables, and he should be able to grasp how these enter in the solution of problems.

Applying knowledge and understanding: The student should be able to apply the forementioned notions to solve medium level problems, and to understand how they relate to concepts seen in different courses.

Making judgements: The student should be able to evaluate coherence and correctness of the proofs he gives during the written test.

Communication skills: The student should be able to communicate in a clear and precise way, suitable for a scientist-to-be in an intermediate stage of his formation.

Lerning skills: The student should be able to link the content of the course with what he learned in the first year.

## Prerequisites

Analyis for functions of one real variable; linear geometry; linear algebra.

Any book of Elements of Mathematical Analysis 2.

## Course contents summary

Normed and metric spaces.

Limits and continuity of functions of several real variables.

Curves.

Differential calculus for functions of several real variables.

Implicit Function Theorem and consequences.

Multiple integrals.

Potential functions and differential forms.

Full proof is provided for most statements.

For Physics students, elements of differential equations

## Course contents

Norms, distances, equivalent norms and equivalent distances.

Limits and continuity of functions of several real variables.

Regular curves, simple curves, equivalences among curves, paths, unit tangent vector to regular paths, curve lenghts, integrals of continuous functions along paths; work of a field along a path..

Differential calculus for functions of several real variables: directional derivatives and their geometric meaning, partial derivatives, gradients, differentiation rules, tangent hyperplanes and their geometric meanings, Schwarz Theorem, Taylor formula, quadratic forms, local maxima and minima.

Implicit Function Theorem, Inverse Function Theorem, smooth surfaces, Lagrange Theorem. Multiple integrals: definitions, reduction theorem, changes of variables. Integral in two and several dimensions. Integral on surfaces.Potential functions and differential forms

Scalar and vector potential in star-shaped, simply connected, general domains. Closed and exact differential forms. Paths and integration of differential forms.

For Physics students: differential equations; existence; uniqueness; separation of variables; linear equations up to second order; notes on qualitative study.

## Recommended readings

Lectures closely follow

E. Acerbi e G. Buttazzo, Secondo corso di analisi matematica. Pitagora Bologna (2016).

One may use instead any good book on analysis in several variables, as e.g.

G. Prodi: Lezioni di Analisi Matematica II. ETS Pisa (1974)

W. Fleming: Functions of several variables. Providence 1965.

Through the platform Elly the students may access texts and solutions of all the previous final exams, and material on some complementary subjects.

Any book of Elements of Mathematical Analysis 2.

## Teaching methods

Lectures are held in the classroom, encompassing both theoretical and applied aspects. Moreover, exercises are solved by students with the guidance of the teacher, so as to verify the degree of comprehension and knowledge of the students.

## Assessment methods and criteria

It is compulsory to book the exam via Esse3.

The final exam consist of a written and an oral session. Students are admitted to the oral sessions only if they pass the written examination. The written examination lasts 2 hours and consists in 3 open questions, each with a maximum score of 10. The students should exhibit calculus skills and mastery of different subjects taught in the course. Marks are given to each question, according to theoretical correctness, precision of execution, precision of exposition. The written examination is passed with a minimum score of 15. The results of the written examination are given through Esse3, generally within 2 days.

The oral examination consists of a discussion about the written examination, and of questions to verify the level of comprehension of the theoretical parts of the course. The score for the oral examination ranges from -15 (serious and diffuse weaknesses on fundamental concepts) to +7 (deep knowledge and fluent exposition, also of minor points in the theory) and is added to the score of the written part. The result is communicated immediately.

## Other informations

It is strongly recommended to attend the lessons.